Mathematical modeling and optimization technique of anticancer antibiotic adsorption onto carbon nanocarriers

This study employs a combination of mathematical derivation and optimization technique to investigate the adsorption of drug molecules on nanocarriers. Specifically, the chemotherapy drugs, fluorouracil, proflavine, and methylene blue, are non-covalently bonded with either a flat graphene sheet or a spherical \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm C}_{60}$$\end{document}C60 fullerene. Mathematical expressions for the interaction energy between an atom and graphene, as well as between an atom and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm C}_{60}$$\end{document}C60 fullerene, are derived. Subsequently, a discrete summation is evaluated for all atoms on the drug molecule utilizing the U-NSGA-III algorithm. The stable configurations’ three-dimensional architectures are presented, accompanied by numerical values for crucial parameters. The results indicate that the nanocarrier’s structure effectively accommodates the atoms on the drug’s carbon planes. The three drug types’ molecules disperse across the graphene surface, whereas only fluorouracil spreads on the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm C}_{60}$$\end{document}C60 surface; proflavine and methylene blue stack vertically to form a layer. Furthermore, all atomic positions of equilibrium configurations for all systems are obtained. This hybrid method, integrating analytical expressions and an optimization process, significantly reduces computational time, representing an initial step in studying the binding of drug molecules on nanocarriers.


Continuous and discrete approximations
We tested the mathematical expressions derived from the continuous approximation of two molecules and the hybrid discrete-continuous approximation.In the continuous approach, the equilibrium interaction energy between two spherical C 60 fullerenes is −9.88243 kcal/mol with the distance between their centers of 10.02325 Å.The equilibrium interaction energy between a spherical C 60 fullerene and an infinite flat graphene sheet is −29.19610kcal/mol, and the distance from the center of the fullerene to the graphene is 6.49861 Å.
In hybrid discrete-continuous approximation, the graphene is assumed to be a continuous plane whereas the fullerene is considered as a discrete structure.The optimal interaction energy between the C 60 fullerene as a discrete molecule and the infinite flat graphene sheet is −30.67018kcal/mol, and a distance from the center of the fullerene to the graphene is 6.37027 Å.Assuming one fullerene molecule to be a 1 perfect spherical structure interacting with another discrete fullerene structure yields a minimum energy of −11.17408 kcal/mol and the distance between their center of 9.59308 Å.Furthermore, for the discrete summation between two C 60 fullerenes, we obtain the energy of −12.06251 kcal/mol with the distance between their centers of 8.76983 Å.
When comparing the interaction between two C 60 fullerenes, each system exhibits an energy difference of 1 kcal/mol, which would decrease with a higher number of atoms in the system.The energies obtained from the two models for the interaction between C 60 and graphene are in good agreement, and the separation distances are approximately equal.

Energy function for optimization
In this appendix, energy equations obtained from the hybrid approach for the system involving fluorouracil molecule C 4 H 3 FN 2 O 2 are given.For the interaction with the flat graphene sheet, we employ the interaction energy between a point and the infinite plane given in (3) and sum over 12 atoms of the fluorouracil, then the total interaction energy between one fluorouracil and graphene is given by where δ represents the distance between an atom and the graphene surface which is the absolute of z-component of each atom in the optimization process, η p is the mean atomic surface density of the graphene, and A M N and B M N denote the attractive and repulsive Lennard-Jones constants, respectively, utilizing the mixing rule of two atomic species M and N .
For the system of two fluorouracil molecules, we use the energy equation (2.1) for each of the fluorouracil molecule and combine with the energy contribution using discrete approach for the interaction between two molecules of fluorouracil.
In the system of C 60 , we employ the total interaction energy between a surface of a sphere and a single atom given in ( 5) to determine the interaction energy between a fluorouracil molecule and a C 60 fullerene which is given by Again the energy of two fluorouracil molecules intercting with the C 60 , the interaction between two fluorouracils is calculated using the discrete summation and equation (2.2) is evaluated for each of the fluorouracil for the total energy of the system.
In the optimization process, we minimize the total energy equation in order to find the stable configuration where all atomic positions are obtained and they are available upon request.

Alignment of drugs based on incline and rotational angles
This appendix shows possible alignments of two drug molecules using the incline angle α and the rotational angle β.There are three configurations which defined as Figure 3.1: Schematic models for alignments of proflavine.

Figure 3 . 2 :
Figure 3.2: Schematic models for alignments of fluorouracil and methylene blue.